Theorem isusp 24149

Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
isusp.1	⊢ 𝐵 = (Base‘𝑊)
isusp.2	⊢ 𝑈 = (UnifSt‘𝑊)
isusp.3	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
isusp	⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3468	. 2 ⊢ (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2		0nep0 5313	. . . . 5 ⊢ ∅ ≠ {∅}
3		isusp.1	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
4		fvprc 6850	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
5	3, 4	eqtrid 2776	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅)
6	5	fveq2d 6862	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7		ust0 24107	. . . . . . . . . 10 ⊢ (UnifOn‘∅) = {{∅}}
8	6, 7	eqtrdi 2780	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
9	8	eleq2d 2814	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10		isusp.2	. . . . . . . . . 10 ⊢ 𝑈 = (UnifSt‘𝑊)
11	10	fvexi 6872	. . . . . . . . 9 ⊢ 𝑈 ∈ V
12	11	elsn 4604	. . . . . . . 8 ⊢ (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
13	9, 12	bitrdi 287	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
14		fvprc 6850	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
15	10, 14	eqtrid 2776	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	eqeq1d 2731	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
17	13, 16	bitrd 279	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
18	17	necon3bbid 2962	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
19	2, 18	mpbiri 258	. . . 4 ⊢ (¬ 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
20	19	con4i 114	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
21	20	adantr 480	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
22		fveq2 6858	. . . . . 6 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
23	22, 10	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
24		fveq2 6858	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
25	24, 3	eqtr4di 2782	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
26	25	fveq2d 6862	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
27	23, 26	eleq12d 2822	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
28		fveq2 6858	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
29		isusp.3	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
30	28, 29	eqtr4di 2782	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
31	23	fveq2d 6862	. . . . 5 ⊢ (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
32	30, 31	eqeq12d 2745	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
33	27, 32	anbi12d 632	. . 3 ⊢ (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
34		df-usp 24145	. . 3 ⊢ UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
35	33, 34	elab2g 3647	. 2 ⊢ (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
36	1, 21, 35	pm5.21nii 378	1 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447  ∅c0 4296  {csn 4589  ‘cfv 6511  Basecbs 17179  TopOpenctopn 17384  UnifOncust 24087  unifTopcutop 24118  UnifStcuss 24141  UnifSpcusp 24142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6464  df-fun 6513  df-fv 6519  df-ust 24088  df-usp 24145

This theorem is referenced by:  ressust  24151  ressusp  24152  tususp  24159  uspreg  24161  ucncn  24172  neipcfilu  24183  ucnextcn  24191  xmsusp  24457